In a hydrogen like atom electron make transition from an energy level with quantum number `n` to another with quantum number `(n - 1)` if `n gtgt1` , the frequency of radiation emitted is proportional to :

Options

A. `(1)/(n^(2))`
B. `(1)/(n^(3//2))`
C. `(1)/(n^(3))`
D. `(1)/(n)`

Answer

To solve the problem, we need to determine how the frequency of radiation emitted during an electron transition in a hydrogen-like atom is related to the principal quantum number \( n \).

### Step-by-Step Solution:

1. **Understanding the Energy Levels**:
In a hydrogen-like atom, the energy of an electron in a given energy level \( n \) is given by the formula:
\[ E_n \propto -\frac{Z^2}{n^2} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

2. **Energy Change During Transition**:
When an electron transitions from an energy level \( n \) to \( n-1 \), the change in energy \( \Delta E \) can be expressed as:
\[ \Delta E = E_n - E_{n-1} \]

3. **Calculating the Energies**:
Using the energy formula, we can write:
\[ E_n \propto -\frac{1}{n^2} \quad \text{and} \quad E_{n-1} \propto -\frac{1}{(n-1)^2} \]

4. **Finding the Change in Energy**:
The change in energy \( \Delta E \) can be calculated as:
\[ \Delta E \propto -\frac{1}{n^2} + \frac{1}{(n-1)^2} \]
Simplifying this expression:
\[ \Delta E \propto \left( \frac{(n-1)^2 - n^2}{n^2(n-1)^2} \right) \]
\[ = \frac{(n^2 - 2n + 1) - n^2}{n^2(n-1)^2} = \frac{-2n + 1}{n^2(n-1)^2} \]

5. **Considering \( n \gg 1 \)**:
Since \( n \) is very large, we can approximate:
\[ \Delta E \approx \frac{-2n}{n^2(n-1)^2} \approx \frac{-2}{n^3} \]

6. **Relating Energy to Frequency**:
The energy emitted during the transition is also related to the frequency \( f \) of the emitted radiation by Planck's equation:
\[ \Delta E = h f \]
Therefore, we can write:
\[ f \propto \Delta E \]

7. **Final Relationship**:
Since we found that \( \Delta E \propto \frac{1}{n^3} \), we can conclude that:
\[ f \propto \frac{1}{n^3} \]

### Conclusion:
The frequency of radiation emitted when an electron transitions from energy level \( n \) to \( n-1 \) in a hydrogen-like atom is proportional to \( \frac{1}{n^3} \).